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Polynomial Functions in Roller Coasters
Transcript of Polynomial Functions in Roller Coasters
So...who are the masterminds behind the math?
We wouldn't have these polynomial functions (or roller coasters for that matter) if we did not have someone to discover and expand on these ideas!
So what exactly is a polyomial function???
It's actually an algebraic expression with one or more terms and usually each of those terms have a certain degree.
They are expressed in a particular form:
a= any real #
coefficient with zero is the constant term
coefficient with the greatest power of x is the leading coefficient
The Persian mathematician, astronomer, and poet
showed how to express roots of cubic equations by line segments, but he could not find a formula for the roots.
In the early 13th century, the great Italian mathematician
achieved a close approximation to the solution of the cubic equation x3 + 2x2 + cx = d.
Early in the 16th century, the Italian mathematicians
Scipione del Ferro
solved the general cubic equation in terms of the constants appearing in the equation. Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree.
An important development in algebra in the 16th century was the introduction of symbols for the unknown and for algebraic powers and operations. The French philosopher and mathematician
, is known for his so-called rule of signs for counting the number of what Descartes called the "true" (positive) and "false" (negative) roots of an equation.
Work continued through the 18th century on the theory of equations, but not until 1799 was the proof published, by the German mathematician
Carl Friedrich Gauss,
showing that every polynomial equation has at least one root in the complex plane.
Polynomial succeeded the term binomial, and was made simply by replacing the Latin root bi- with the Greek poly-, which comes from the Greek word for many. The word polynomial was first used in the 17th century.
The Alexandrian mathematicians
Hero of Alexandria
expanded on the ancient knowledge of solutions of equations and in turn found a home early in the Islamic world, where it was known as the "science of restoration and balancing."
The Egyptian mathematician
had stated and proved the basic laws and identities of algebra and solved such complicated problems as finding x, y, and z such that x + y + z = 10, x^2 + y^2 = z^2, and xz = y^2.
Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about arbitrarily high powers of the unknown x, and work out the basic algebra of polynomials. This included the ability to multiply, divide, and find square roots of polynomials as well as a knowledge of the binomial theorem.
For this example, this is the first equation used:
-(x ) + (3.5x ) - (2.5x ) - (12.5x )+ 1.5x + 9
We tried again with a different function to get this graph:
x +4x - 4x -26x -7x +28x +20x
The last equation we made up is for this graph:
-2x -13x + 26x -7x + 28x +20x
Looking at these
3 graphs; if they
were all made into
which would you ride?
So...How can we make our OWN roller coasters?
What we decided to do is to create some functions with random coefficients and different degrees and orientations to see what we would come up with! We ended up with 3 different examples or "roller coasters"...
Welcome, mathematicians, to the Polynomial Power Function Coaster! We ask that you would remain seated and keep all hands and feet inside the car at all times. If motion sickness makes you uneasy we suggest you leave the queue now before you puke on the others trying to have fun. Let's crunch some numbers and hope you enjoy the ride! :)
Of course, we understand that more complex procedures go into actually building roller coasters.
However, polynomial functions are not limited to just roller coasters.
Polynomials can also have rational and complex coefficients as well as real and integers!
This means you can find these handy functions just about anywhere!
Polynomial functions can appear constant, linear, quadratic, cubic, quartic, and so on
Whole number powers of x, known as
Real number coeficients of
Speciefied/differentiating values for
is used to determine the general shape of the graph and the rest of the equation, is used in relvance to the maxima and minima, the size & actual shape, and the following curves/turning points
How exactly do polynomial functions relate and contribute to the shape of a rollercoaster?
To determine the general shape of the graph (or rollercoaster for this case) of a polynomial function:
1. Use leading-term test to determine end behavior
2. Find the zeros of the function
3. Use the x-int. to divide the x-axis into intervals, then choose a 'test' point to determine the sign
4. Find the y-int.
5. Find any additional function values to determine general shape
Lets give it a try!
Consider the example:
h(x) = x^3 + 3x^2 - x - 3
Determine the general shape of the of this polynomial function using the process descibed above
Is this what you got?
Reflections / Conclusions
Thank you for riding the Polynomial Power Function Coaster! We hope you enjoyed the ride and learned something today! Have a great day!